In this paper, we develop an accurate and efficient Nyström volume integral equation (VIE) method for the Maxwell equations for large number of 3-D scatterers. The Cauchy Principal Values that arise from the VIE are computed accurately using a finite size exclusion volume together with explicit correction integrals consisting of removable singularities. Also, the hyper-singular integrals are computed using interpolated quadrature formulae with tensor-product quadrature nodes for several objects, such as cubes and spheres, that are frequently encountered in the design of meta-materials . The resulting Nyström VIE method is shown to have high accuracy with a minimum number of collocation points and demonstrate p-convergence for computing the electromagnetic scattering of these objects. Numerical calculations of multiple scatterers of cubic and spherical shapes validate the efficiency and accuracy of the proposed method.
In this paper, we present accurate and economic integration quadratures for hypersingular functions over three simple geometric shapes in ℝ3 (spheres, cubes, and cylinders). The quadrature nodes are made of the tensor-product of 1-D Gauss nodes on [–1,1] for non-periodic variables or uniform nodes on [0,2π] or [0,π] for periodic ones. The quadrature weights are converted from a brute-force integration of the hypersingular function through interpolating the smooth component of the integrand. Numerical results are presented to validate the accuracy and efficiency of computing hypersingular integrals, as in the computations of Cauchy principal values, with a minimum number of quadrature nodes. The pre-calculated quadrature tables can be then readily used to implement Nyström collocation methods of hypersingular volume integral equations such as the one for Maxwell equations.
The method-of-moments implementation of the electric-field integral equation yields many code-verification challenges due to the different sources of numerical error and their possible interactions. Matters are further complicated by singular integrals, which arise from the presence of a Green's function. In this paper, we provide approaches to separately assess the numerical errors arising from the use of basis functions to approximate the solution and the use of quadrature to approximate the integration. Through these approaches, we are able to verify the code and compare the error from different quadrature options.
Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules.
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